Harvey Friedman, Classically and intuitionistically provably recursive functions, Higher Set Theory (D.S. Scott and G.H. Muller, eds.), Springer-Verlag LNM 699, 1978, pp. 21--27.  Home/Search   Document Not in Database   Summary   Related Articles   Check

...., is given by: j(x) x At least for first order arithmetic, this means that a sentence # is true , precisely when its Friedman translation (#) U is true in . Introduce an extra propositional constant U to the language, interpreted by the arrow 1 # The Friedman translation (see [1]) puts U for atomic formulas P , and commutes with all (first order) logical operations. Since is (standard, Kleene) realizability over Set # and the latter is Kripke forcing over Set, one has: # Most of the research for this paper was carried out during the PIONIER project The Geometry of ....

H.M. Friedman. Classically and Intuitionistically Provably Recursive Functions. In Muller and Scott, editors, Higher Set Theory, pages 21--27. Springer-Verlag, 1978. 10

....is heavily impredicative, as it involves universal quanti cation over all bad sequences. Thus, it is natural, and as it turns out, quite challenging, to ask whether it is possible to give a constructive (and predicative) proof of Higman s lemma. In a remarkable (and short) paper, Friedman [15] introduces a new and simple technique, the A translation, which enables him to give simple proofs of the fact that rst order classical Peano arithmetic and classical higher order arithmetic are conservative over their respective intuitionistic version over 2 sentences. His technique also ....

Friedman, H. Classically and intuitionistically provably recursive functions. Higher set theory (G.H. Muller and Dana S. Scott, editors), Lecture Notes in Mathematics, Vol. 699, Springer-Verlag, Berlin (1978), 21-28.

....our setting, this would correspond to a proof of # #A# where is the translation of A. It is more economical and allows some further applications if we instead parameterize the translation by a propositional parameter p and verify that ##A# p =# p#, an idea that goes back to Friedman [12] and was also employed by Lamarche [17] in the linear setting. It is convenient to introduce the parametric negation p A = A#p, where A is the usual negation in JILL. Since the translation of A becomes a linear hypothesis, all connectives except# , 1, and can simply be dualized. For the ....

Harvey Friedman. Classically and intuitionistically provably recursive functions. In D.S. Scott and G.H. Muller, editors, Higher Set Theory, pages 21--27. Springer-Verlag LNM 699, 1978.

....0 (x, y) # HA # #x#yA 0 (x, y) We still need, however, an algorithm for transforming a proof HA # #yA 0 (x, y) into a proof HA # #yA 0 (x, y) The A translation 11 is a simple technique for performing such translation which applies to many intuitionistic theories. Definition 2. 9 ([Fri78]) Let A # L(HA) With each formula F # L(HA) such that the free variables of A are not bounded in F ) associate a formula (F ) A # L(HA) called the A translation of F , in the following way: F ) A results when all prime formulas P in F are replaced by P # A. Theorem 2.10 (Soundness ....

....The following rule holds, HA # #x#yA 0 (x, y) alg =# HA # #x#yA 0 (x, y) for A 0 quantifier free. 10 In arithmetical systems like HA any quantifier free formulas A0(x) is equivalent to a prime formula t(x) 0, for a suitable term t. 11 The A translation was developed by H. Friedman [Fri78] and independently investigated by Dragalin [Dra79] Variants of the translation were considered in [Lei85] and [Tv88] 10 Proof: In HA any quantifier free formula A 0 (x, y) can be written as a prime formula P (x, y) see footnote 10) In this way HA # #x#yA 0 (x, y) implies, HA # ( #yP ....

H. Friedman. Classically and intuitionistically provably recursive functions. In D. Scott and G. Mller, editors, Higher Set Theory, Lecture Notes in Mathematics, volume 669, pages 21--28. Springer Verlag, 1978.

.... losing grip on computational aspects, b) extracted programs will contain less junk, as demonstrated in [6] where a short and ecient normalisation program for the simply typed lambda calculus is extracted, c) classical logic can be treated more smoothly [7] avoiding expensive proof translations [27, 12, 13, 14], and (d) crucial axioms of analysis like the classical quanti er free axiom of choice will be realisable (this is inspired by [4] In this connection it will be interesting to compare our work with the approach in [35] to program development in analysis via G odel s Dialectica Interpretation, ....

H. Friedman. Classically and intuitionistically provably recursive functions. In: D.S. Scott and G. H. Muller, editors, Higher Set Theory, LNM 669, 21-28, 1978.

....and our goal is given by goal formulas. For quantifier free formulas this clearly can always be achieved by inserting double negations in front of every atom (cf. the definitions of definite and goal formulas) This corresponds to the original (unrefined) so called A translation of Friedman [11] (or Leivant [15] However, in order to obtain reasonable programs which do not unneccessarily use higher types or case analysis we want to insert double negations only at as few places as possible. We describe a more economical general way to obtain definite and goal formulas, following [2, 3] ....

Harvey Friedman. Classically and intuitionistically provably recursive functions. In D.S. Scott and G.H. Muller, editors, Higher Set Theory, volume 22 669 of Lecture Notes in Mathematics, pages 21--28. Springer Verlag, Berlin, Heidelberg, New York, 1978.

....from classical proofs A classical, i.e. non constructive proof of an existential statement can under certain circumstances be translated into a constructive proof, and hence yields an algorithm. For background we refer to [5] where a refinement of the A translation going back to work of Friedman [9] and Leivant [14] is described. 6.3.1 classical proof to constr proof (classical proof to constr proof proof) transforms a classical proof of an 89 Gammastatement to a constructive one. The user is allowed to use free or global Pi 0 1 Gammaassumptions, i.e. the goal should be of the form ....

Harvey Friedman. Classically and intuitionistically provably recursive functions. In D.S. Scott and G.H. Muller, editors, Higher Set Theory, volume 669 of Lecture Notes in Mathematics, pages 21--28. Springer Verlag, Berlin, Heidelberg, New York, 1978.

.... proofs under certain circumstances can be translated into constructive proofs, and hence yield algorithms via program extraction (cf. section 5) There is a substantial literature on that subject, and the MINLOG system supports a variant, known as A translation , which goes back to work of Friedman (1978) and Leivant (1985) We will explain this translation by means of an example concerning minimization on finite binary trees. We will also use this example to discuss a second technique for program development: the pruning operation going back to Goad (1980) 6.1. Search through binary trees In ....

....translated into a formula which follows constructively from ax i . Of course, this modified translation (i.e. s translation followed by the replacement A) also affects the formula A 0 , but still transforms the formula s A 0 s into a provable formula. Remarks: 1. Friedman s original translation (Friedman, 1978) replaces every atomic formula R by R A and not, as we did, by R A A. But clearly the formulas R A and R A A are constructively equivalent assuming decidability of R. We have chosen the latter variant, since in MINLOG we prefer reasoning with implications rather with disjunctions. 2. Another way ....

Friedman, H.: 1978, `Classically and intuitionistically provably recursive functions'. In: D. Scott and G. (eds.): Higher Set Theory, Vol. 669 of Lecture Notes in Mathematics. pp. 21--28.

....assumptions are definite formulas and our goal is given by goal formulas. Clearly this can always be achieved by inserting double negations in front of every atom (cf. the definitions of definite and goal formulas) This corresponds to the original (unrefined) so called A translation of Friedman [7] (or Leivant [9] However, in order to obtain reasonable programs which do not unneccessarily use higher types or case analysis we want to insert double negations only at as few places as possible. We describe a general way to obtain definite and goal formulas, following [1, 2] It consists in ....

Harvey Friedman. Classically and intuitionistically provably recursive functions. In D.S. Scott and G.H. Muller, editors, Higher Set Theory, volume 669 of Lecture Notes in Mathematics, pages 21--28. Springer Verlag, Berlin, Heidelberg, New York, 1978.

....a = 0 in R=P as z(a) By using the simple trick to prove z(b l )z(a i ) rather than z(b l ) by induction, we avoid the use of contradiction and the proof is reformulated to a more positive proof which uses only z(x) and not its negation. This trick is used in Friedman s A translation [Fri78] which translates classical proofs in Peano arithmetic into constructive proofs. Theorem 7. Whenever fg = 1, where f = amX m Delta Delta Delta a 0 2 R[X ] and g = b n X n Delta Delta Delta b 0 , then z(a m ) z(a 1 ) holds for any zero function z. Proof. We prove the ....

H. Friedman. Classically and intuitionistically provably recursive functions. In D. Scott and G. Muller, editors, Higher Set Theory, volume 669 of Lecture Notes in Mathematics, pages 21--28. Springer-Verlag, 1978.

....replacing every atomic formula by ( and replacing by . In the case when is this was rst used in (Prawitz and Malmn as 1968) to interprete intuitionistic The Russell Prawitz Modality 7 logic in minimal logic. More recently the general translation has been introduced by Friedman, (Friedman 1978) (and also by Dragalin) as a useful tool in connection with proving closure under Markov s rule. See also (Leivant 1985; Murthy 1991) We have the following connection with the J 3 translation. Proposition 10. If T is a standard theory for L and is any formula such that T ( then ....

Friedman, H. (1978) Classically and intuitionistically provably recursive functions. In G.H.

....We will concentrate on the question of classical versus constructive proofs. It is well known that any classical proof of a specification of the form 8x9yB with B quantifier free can be transformed into a constructive proof of the same formula (for particularly simple proofs, cf. Friedman [5] or Leivant [7] However, when it comes to extraction of a program from a proof obtained in this way, one easily ends up with a mess. Therefore, some refinements of the standard transformation are necessary. In the present paper we make use of a refined method of extracting programs from ....

Harvey Friedman. Classically and intuitionistically provably recursive functions. In D.S. Scott and G.H. Muller, editors, Higher Set Theory, volume 669 of Lecture Notes in Mathematics, pages 21--28. Springer Verlag, Berlin, Heidelberg, New York, 1978.

....assumptions are definite formulas and our goal is given by goal formulas. Clearly this can always be achieved by inserting double negations in front of every atom (cf. the definitions of definite and goal formulas) This corresponds to the original (unrefined) so called A translation of Friedman [11]. However, in order to obtain reasonable programs which do not unneccessarily use higher types we want to insert double negations only at as few places as possible. We describe a general way to obtain definite and goal formulas, following [5] It consists in singling out some predicate symbols as ....

....is considerably more efficient than the brute force program systematically testing all pairs i j. This contrasts the program extracted by Murthy [14] from the translation of the classical Nash Williams proof of (a specialization of) Higman s lemma using the original Friedman translation [11]. The proof will be given informally, but in such a detail that it will be immediately clear how to formalize it. We sketch how this proof is transformed into a constructive one by applying the method above, and discuss the extracted program. Lemma 3.10. 8f; g9i; j:i j f(j) 6 f(i) g(j) 6 ....

Harvey Friedman. Classically and intuitionistically provably recursive functions. In D.S. Scott and G.H. Muller, editors, Higher Set Theory, volume 669 of Lecture Notes in Mathematics, pages 21--28. Springer Verlag, Berlin, Heidelberg, New York, 1978.

....typing of programs, people have been attempting to derive programs from classical proofs. Murthy [27, 28] has shown that the C.P.S. transform is nothing other than the Kolgomorov double negation embedding of classical logic into intuitionistic logic. If one uses Friedman s A translation [12] to replace all instances of with a fresh proposition A, one has an embedding of intuitionistic logic into minimal logic. One can then extract programs from classical proofs, in much the same way as one would extract programs from intuitionistic proofs, but translating double negation ....

H. Friedman. Classically and intuitionistically provably recursive functions. In D. S. Scott and G. H. Muller, editors, Higher Set Theory, pages 21--28. Springer-Verlag, 1978. Lecture Notes in Mathematics 699.

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Harvey Friedman, Classically and intuitionistically provably recursive functions, Higher Set Theory (D.S. Scott and G.H. Muller, eds.), Springer-Verlag LNM 699, 1978, pp. 21--27.

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Harvey Friedman, Classically and intuitionistically provably recursive functions, Higher Set Theory (D.S. Scott and G.H. Muller, eds.), Springer-Verlag LNM 699, 1978, pp. 21--27.

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Harvey Friedman, Classically and intuitionistically provably recursive functions, Higher set theory (D.S. Scott and G.H. Muller, editors), Springer-Verlag LNM 699, 1978, pp. 21--27.

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Harvey Friedman, Classically and intuitionistically provably recursive functions, Higher Set Theory (D.S. Scott and G.H. Muller, eds.), Springer-Verlag LNM 699, 1978, pp. 21--27.

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H. Friedman, Classically and intuitionistically provably recursive functions, in Higher Set Theory: proceedings, Oberwolfach, Germany, April 12-23, 1977.

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H. Friedman. Classically and intuitionistically provably recursive functions. In D. Scott and G. Muller, editors, Higher Set Theory, volume 669 of Lecture Notes in Mathematics, pages 21--28. Springer, Berlin, 1978.

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Harvey Friedman. Classically and intuitionistically provably recursive functions. In D. S. Scott and G. H. Muller, editors, Higher Set Theory, volume 699 of Lecture Notes in Mathematics, pages 21--28. Springer-Verlag, 1978.

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Harvey Friedman, Classically and intuitionistically provably recursive functions, Higher Set Theory (D.S. Scott and G.H. Muller, eds.), Springer-Verlag LNM 699, 1978, pp. 21--27.

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Harvey Friedman, Classically and intuitionistically provably recursive functions, Higher set theory (D.S. Scott and G.H. Muller, editors), Springer-Verlag LNM 699, 1978, pp. 21--27.

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H. Friedman. Classically and intuitionistically provably recursive functions. In: Higher set theory. Springer Lect. Notes in Math., n # 669, 21-27 (1977).

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Harvey Friedman. Classically and intuitionistically provably recursive functions. In D.S. Scott and G.H. Muller, editors, Higher Set Theory, volume 669 of Lecture Notes in Mathematics, pages 21--28. Springer Verlag, Berlin, Heidelberg, New York, 1978.

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